Solving Inequalities Flashcards

All 10 cards in this deck

What are the steps to solve a linear inequality with brackets and unknowns on both sides?
e.g. $5(x-3) > 2(x+3)$

1. Expand both sides: $5x - 15 > 2x + 6$
2. Collect the unknowns on one side: $3x > 21$
3. Divide by the coefficient: $x > 7$

What must you do to an inequality sign when you multiply or divide both sides by a negative number?
e.g. $-2x < 6$

Reverse it.
$-2x < 6$ gives $x > -3$

What are the steps to form and solve an inequality from a context?
e.g. a rectangle with sides $x$ cm and $(x+2)$ cm has perimeter less than $20$ cm

1. Form the inequality: $4x + 4 < 20$
2. Solve it: $x < 4$
3. Add any restriction from the context: lengths are positive, so $0 < x < 4$

True or false? The solution $x \geqslant 5$ includes the value $5$ itself.

True.
$\geqslant$ means "greater than or equal to", so the boundary value is included.

What is the solution of $x^2 < k$ (for $k > 0$)?
e.g. $x^2 < 9$

$-\sqrt{k} < x < \sqrt{k}$
So $x^2 < 9$ gives $-3 < x < 3$.

What is the solution of $x^2 \geqslant k$ (for $k > 0$)?
e.g. $x^2 \geqslant 25$

$x \leqslant -\sqrt{k}$ or $x \geqslant \sqrt{k}$
So $x^2 \geqslant 25$ gives $x \leqslant -5$ or $x \geqslant 5$.

What are the steps to solve a quadratic inequality by factorising?
e.g. $x^2 - x - 6 > 0$

1. Factorise and find the critical values: $(x-3)(x+2)=0$, so $x = -2$ and $x = 3$
2. Sketch the U-shaped curve crossing at those values
3. Read off where the curve is above the axis: $x < -2$ or $x > 3$

What are the steps to solve an inequality containing an algebraic fraction?
e.g. $3x + \dfrac{8}{x} \leqslant 10$ where $x > 0$

1. Rearrange into quadratic form: $\dfrac{3x^2-10x+8}{x} \leqslant 0$, i.e. $3x^2 - 10x + 8 \leqslant 0$
2. Factorise: $(3x-4)(x-2) \leqslant 0$, critical values $\frac{4}{3}$ and $2$
3. Choose the region: $\frac{4}{3} \leqslant x \leqslant 2$

What are the steps to list the integer values satisfying a quadratic inequality?
e.g. $x^2 < 9$

1. Solve the inequality: $-3 < x < 3$
2. Check whether the end values are included: here they are not
3. List every integer inside: $-2, -1, 0, 1, 2$

True or false? The solution "$x < -4$ or $x > 5$" may be written as the single statement $-4 > x > 5$.

False.
Embedded inequalities like that are not accepted as a final answer — write two separate inequalities.