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Equations, inequalities and systems

Solving an equation means finding the values of the unknown that make it true. This topic reviews linear and quadratic equations, factoring, equations with fractions and roots (and why you must check), inequalities with a sign chart and systems of linear equations by elimination.

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Linear and quadratic equations

An equation becomes an equivalent one by doing the same thing to both sides: adding, subtracting, multiplying or dividing by something other than zero. A linear one ( $a x + b = 0$ ) is solved directly: $x = - b / a$ . A quadratic $a x^{2} + b x + c = 0$ is solved with the quadratic formula:

x = \frac{- b \pm Δ}{2 a}, Δ = b^{2} - 4 a c

The discriminant

Δ

tells how many times the parabola cuts the

x

axis.

If the roots $r_{1}$ and $r_{2}$ are known, the polynomial factors as $a (x - r_{1}) (x - r_{2})$ . Conversely, if a product is zero, one of the factors is zero: that is why factoring is a way of solving.

Fractions and roots: always check

With unknowns in the denominator, multiply by the common denominator; but the values that make it zero cannot be solutions. With a root, isolate it and square both sides; squaring can create extraneous solutions (from $a = b$ we go to $a^{2} = b^{2}$ , which $a = - b$ also satisfies). That is why each candidate is substituted into the original equation.

Inequalities

To solve $f (x) > 0$ (or $<$ , $\geq$ , $\leq$ ) bring everything to one side, find the critical points (where $f$ is 0 or does not exist) and test the sign of $f$ at one value of each interval: that is the sign chart. When an inequality is multiplied or divided by a negative number, it flips.

Systems of linear equations

They are written as an augmented matrix $[A ∣ b]$ and row operations are applied (swapping rows, multiplying a row by a nonzero number, adding a multiple of one row to another) until it is in echelon form: that is Gaussian elimination (or Gauss-Jordan if it reaches the identity). A system can have one solution, none or infinitely many:

With two unknowns, each equation is a line and the solution is their intersection.

After elimination…	Type	Solutions
one pivot per unknown	consistent, independent	one
fewer pivots than unknowns, no $0 = c$ rows	consistent, dependent	infinitely many (free variables)
a $0 = c$ row with $c \neq = 0$	inconsistent	none

The calculator solves systems by the method you choose (Gauss-Jordan, Gauss, Cramer, inverse matrix, substitution, equating or elimination) and shows each row operation. In equations it checks each solution before giving it.

Common mistakes

Dividing by $x$ to simplify: the solution $x = 0$ is lost. Better to factor it out.
Forgetting the $\pm$ when taking a square root: $x^{2} = 9$ gives $x = 3$ and $x = - 3$ .
Not checking the solutions of an equation with roots or fractions.
Multiplying an inequality by an expression that can be negative without flipping it (better to use the sign chart).

Worked examples

Example 1The quadratic formula

Solve

x^{2} + x - 1 = 0

Solution

$a = 1$ , $b = 1$ , $c = - 1$ : $Δ = 1^{2} - 4 (1) (- 1) = 5 > 0$ , two real roots.
$x = \frac{- 1 \pm 5}{2}$ , that is, $x \approx 0.618$ and $x \approx - 1.618$ .

x = \frac{- 1 - 5}{2}, x = \frac{- 1 + 5}{2}

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Equations and inequalities: type x^2+x-1=0 and press Compute. The Answer is $x = \frac{- 5 - 1}{2}, x = \frac{5 - 1}{2}$ .
The steps identify $a$ , $b$ and $c$ , expand the discriminant and substitute into the formula.

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Example 2Negative discriminant

Solve

x^{2} + 4 = 0

Solution

$Δ = 0 - 4 (1) (4) = - 16 < 0$ : the parabola $y = x^{2} + 4$ does not cut the axis, there are no real solutions.
In the complex numbers: $x = \pm \frac{16}{2} i = \pm 2 i$ .

No real solutions; complex roots

\pm 2 i

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Equations and inequalities: type x^2+4=0 and press Compute. The Answer is «No real solutions», with the complex roots $2 i$ and $- 2 i$ .

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Example 3Factoring a trinomial

Factor

6 x^{2} + x - 2

Solution

Roots: $Δ = 1 + 48 = 49$ , $x = \frac{- 1 \pm 7}{12}$ → $x = \frac{1}{2}$ and $x = - \frac{2}{3}$ .
$6 (x - \frac{1}{2}) (x + \frac{2}{3}) = (2 x - 1) (3 x + 2)$ (the 6 is split as $2 \cdot 3$ ).
Check: $(2 x - 1) (3 x + 2) = 6 x^{2} + 4 x - 3 x - 2 = 6 x^{2} + x - 2$ . ✓

(2 x - 1) (3 x + 2)

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Factor: type 6x^2+x-2 and press Compute. The Answer is $(2 x - 1) (3 x + 2)$ .

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Example 4An equation with a root

Solve

x + 3 = x - 3

Solution

Square both sides: $x + 3 = (x - 3)^{2} = x^{2} - 6 x + 9$ , that is, $x^{2} - 7 x + 6 = 0$ → $x = 1$ or $x = 6$ .
Check: with $x = 6$ , $9 = 3$ and $6 - 3 = 3$ ✓. With $x = 1$ , $4 = 2$ but $1 - 3 = - 2$ ✗: it is extraneous (it satisfies $x + 3 = - (x - 3)$ ).

x = 6

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Equations and inequalities: type sqrt(x+3)=x-3 and press Compute. The Answer is $x = 6$ .
The steps square, solve the quadratic and discard $x = 1$ when checking.

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Example 5An equation with fractions

Solve

\frac{1}{x} + \frac{1}{x + 1} = 1

Solution

Multiply by $x (x + 1)$ , with $x \neq = 0$ and $x \neq = - 1$ : $(x + 1) + x = x^{2} + x$ , that is, $x^{2} - x - 1 = 0$ .
$x = \frac{1 \pm 5}{2}$ ; neither is 0 or $- 1$ , so both are valid.

x = \frac{1 - 5}{2}, x = \frac{1 + 5}{2}

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Equations and inequalities: type 1/x+1/(x+1)=1 and press Compute. The Answer is $x = \frac{- 5 + 1}{2}, x = \frac{5 + 1}{2}$ .

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Example 6Quadratic inequality

Solve

x^{2} - 4 > 0

Solution

Critical points: $x^{2} - 4 = 0$ → $x = - 2$ and $x = 2$ . They split the line into three intervals.
Sign: at $x = - 3$ , $5 > 0$ ; at $x = 0$ , $- 4 < 0$ ; at $x = 3$ , $5 > 0$ .
It holds outside $[- 2, 2]$ ; the endpoints are excluded because the inequality is strict.

x \in (- \infty, - 2) \cup (2, \infty)

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation Equations and inequalities: type x^2-4>0 and press Compute. The Answer is $x \in (- \infty, - 2) \cup (2, \infty)$ , with the sign chart in the steps.

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Example 7A system of three equations

Solve

x + y + z = 6

2 x - y + z = 3

x + 2 y - z = 2

Solution

Augmented matrix and elimination: $F_{2} \to F_{2} - 2 F_{1}$ gives $- 3 y - z = - 9$ ; $F_{3} \to F_{3} - F_{1}$ gives $y - 2 z = - 4$ .
From the second, $y = 2 z - 4$ ; into the first: $- 3 (2 z - 4) - z = - 9$ → $- 7 z = - 21$ → $z = 3$ , $y = 2$ .
$x = 6 - y - z = 1$ . Check in the third: $1 + 4 - 3 = 2$ ✓.

x = 1, y = 2, z = 3

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation System of equations: type each equation in its box and press Compute. The Answer is $x = 1, y = 2, z = 3$ .
The steps show the augmented matrix and each Gauss-Jordan row operation; the method selector also shows it by Cramer or by substitution.

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Example 8Infinitely many solutions

Solve

x + y = 2

2 x + 2 y = 4

Solution

$F_{2} \to F_{2} - 2 F_{1}$ leaves the row $0 = 0$ : the second equation is the first times 2 (the same line).
One equation with two unknowns remains: one free variable. With $y = t$ : $x = 2 - t$ , for any real $t$ .
If the right side of the second were 6 (that is, $x + y = 3$ ), it would leave $0 = 2$ : parallel lines, an inconsistent system.

x = 2 - t

y = t

t \in R

(consistent, dependent).

Solve it in Simuveliq

In Math → Symbolic calculator, topic Algebra, operation System of equations: type the two equations and press Compute. The Answer is $x = 2 - t, y = t, t \in R$ .
The next example in the list, $x + y = 2$ and $x + y = 3$ , gives «The system has no solution (inconsistent)».

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Equations, inequalities and systems

Linear and quadratic equations

Fractions and roots: always check

Inequalities

Systems of linear equations

Common mistakes

Worked examples

Example 1The quadratic formula

Solution

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Example 2Negative discriminant

Solution

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Example 3Factoring a trinomial

Solution

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Example 4An equation with a root

Solution

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Example 5An equation with fractions

Solution

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Example 6Quadratic inequality

Solution

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Example 7A system of three equations

Solution

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Example 8Infinitely many solutions

Solution

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